平稳过程采样定理的一个注记

平稳过程采样定理是随机过程中的一个重要定理.其将平稳过程离散表示的思想方法在理论上有重要意义,在实际工作中有广泛的应用.文章在证明了一个变差理论中的一个重要引理的基础上,利用谱分解的方法,证明了具有有界谱的平稳随机过程,在任何有限的时间区间内可以被离散地表示,不仅是在均方收敛意义下成立,而且这种均方收敛是关于时间t是一致地成立的.

A note on the sampling theorem of stationary process

The sampling theorem is an important theorem in the stationary process,and discretization of the stationary process has theoretical and practical significance. In this paper a lemma in approximate theory is proved first, then by using the spectral decomposition method,it is proved that the stationary process with bounded spectrum can be discretized if time is limited to a finite closed interval,which is not only in square mean convergence but also in square mean convergence uniformly.